Hyperbolic systems of conservation laws. The theory of classical and nonclassical shock waves.

*(English)*Zbl 1019.35001
Lectures in Mathematics, ETH Zürich. Basel: Birkhäuser. x, 294 p. (2002).

Given its origin, it is not surprising that this book is more a text in style than a research monograph. Nonetheless, it contains a considerable amount of information which will be new to many readers in the area.

The theme of the book is the well-posedness of the Cauchy problem for systems of conservation laws. The systems considered are hyperbolic (a strictly convex entropy density is typically assumed), and restricted to one space variable. This is hardly surprising, as the book articulates recent work in this area based on application of wave front-tracking algorithms, using estimates obtained for the solutions of corresponding Riemann problems. In this context, it is no surprise that the usual assumptions on the variation or oscillation of the initial data are made.

The real thrust of the book, however, is in proving the well-posedness of the Cauchy problem in the class of “nonclassical” solutions. These are weak solutions containing “nonclassical”, undercompressive shocks of strength exceeding a predetermined threshold.

For systems with characteristic fields which are neither genuinely nonlinear nor linearly degenerate, such solutions are an alternative to the well-known “classical” solutions containing only Lax shocks. Applications are given motivating the choice of such a “nonclassical” solution.

An admissibility condition for the “nonclassical” shocks is formulated as a quantitative condition on the corresponding entropy dissipation. Specifically, for admissible shocks the rate of entropy dissipation is equal to the value of a “kinetic function” determined from travelling wave analysis of systems regularized with both diffusion and dispersion.

The theme of the book is the well-posedness of the Cauchy problem for systems of conservation laws. The systems considered are hyperbolic (a strictly convex entropy density is typically assumed), and restricted to one space variable. This is hardly surprising, as the book articulates recent work in this area based on application of wave front-tracking algorithms, using estimates obtained for the solutions of corresponding Riemann problems. In this context, it is no surprise that the usual assumptions on the variation or oscillation of the initial data are made.

The real thrust of the book, however, is in proving the well-posedness of the Cauchy problem in the class of “nonclassical” solutions. These are weak solutions containing “nonclassical”, undercompressive shocks of strength exceeding a predetermined threshold.

For systems with characteristic fields which are neither genuinely nonlinear nor linearly degenerate, such solutions are an alternative to the well-known “classical” solutions containing only Lax shocks. Applications are given motivating the choice of such a “nonclassical” solution.

An admissibility condition for the “nonclassical” shocks is formulated as a quantitative condition on the corresponding entropy dissipation. Specifically, for admissible shocks the rate of entropy dissipation is equal to the value of a “kinetic function” determined from travelling wave analysis of systems regularized with both diffusion and dispersion.

Reviewer: Michael Sever (Jerusalem)

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35L65 | Hyperbolic conservation laws |

35L67 | Shocks and singularities for hyperbolic equations |